2. Polynomial Functions
  3. Characteristics of Polynomial Functions

Exercises


  1. Consider the following polynomial functions.
    1. \( y = -2x^3 + 4x - 5 \)
    2. \( f(x) = 5x^4 + 2x^3 - 4x^2 + x - 7 \)
    3. \( g(x) = x^5 + 2x^3 - 5x + 8 \)
    For each one, perform the following tasks.
    1. Describe the end behaviour of the function.
    2. Determine the maximum and minimum number of turning points.
    3. Determine the maximum and minimum number of \( x \)-intercepts.
  2. Given the graph of the polynomial function \( y = f(x) \), identify the minimum possible degree of the function and the sign of the leading coefficient.
    1. Even degree function, two zeros of order 1, three turning points
    2. Odd degree function, three zeros of order 1, two turning points
    3. Even degree function, six zeros of order 1, five turning points
    4. Odd degree function, two zeros of order 1, three turning points
  3. Determine whether the following statements are true or false. If false, explain or provide a counter-example.
    1. An even degree polynomial must have an even number of turning points.
    2. An odd degree polynomial must have at least one \( x \)-intercept.
    3. An even degree polynomial function has an absolute maximum or absolute minimum value.
    4. A polynomial function with three \( x \)-intercepts must be of degree three or greater.
    5. A polynomial function with \( n \) turning points must be of degree \( n, n \in \mathbb{N} \).
    6. An odd degree polynomial function can have even symmetry (symmetry about the \( y \)-axis ).
  4. Match the equations of the polynomial functions to the most appropriate graph. Justify your choice.
    1. \( y = -x^3 + 2x^2 + 2x + 3 \)
    2. \( y = x^4 - x^3 - 3x^2 + 2x + 3 \)
    3. \( y = x^4 + 5x^3 + 6x^2 - 4x - 3 \)
    4. \( y = x^3 - 3x^2 + 5x - 3 \)
    5. \( y = -2x^6 + 13x^4 - 15x^2 - x + 3 \)
    6. \( y = -2x^5 - 7x^4 + 3x^3 + 18x^2 + 3 \)
    I. Has six x-intercepts
    II. Moves from second to first quadrant,one zero of order 2,three turning points
    III. Moves from third quadrant to first quadrant,one distinct zero
    IV. Moves from second quadrant to fourth quadrant,one distinct zero, two turning points
    V. Moves from second to fourth quadrant,three x-intercepts,four turning points
    VI. Moves from second to first quadrant,two distinct zeros,one turning point,one inflection point
  5. What possible number of zeros can a quartic polynomial function have? Sketch a quartic polynomial function with a negative leading coefficient for each possibility?
  6. Sketch a graph of a polynomial function that satisfies each set of conditions.
    1. Degree three, two distinct \( x \)-intercepts, two turning points, and end behaviour such that \( y \rightarrow \infty \) as \( x \rightarrow -\infty \) and \( y \rightarrow -\infty \) as \( x \rightarrow \infty \)
    2. Degree four, two distinct \( x \)-intercepts, three turning points, and end behaviour such that \( y \rightarrow \infty \) as \( x \rightarrow \pm \infty \)
    3. Degree four, negative leading coefficient, three distinct \( x \)-intercepts, three turning points
    4. Degree three, positive leading coefficient, one \( x \)-intercept, two turning points
    5. Degree five, negative leading coefficient, two distinct \( x \)-intercepts, two turning points
    6. Degree five, positive leading coefficient, one \( x \)-intercept, four turning points
  7. Sketch a possible graph of each function by identifying the end behaviours and determining the \( x \)- and \( y \)-intercepts of the function.
    1. \( f(x) = (x - 1)(x - 3)(x + 1)(x + 4) \)
    2. \( y = -2x^3 - 3x^2 + 9x \)
  8. Graph of x^3 + x^2 - 6x Given the graph of the polynomial function \( f(x) = x^3 + x^2 - 6x \), sketch the graph of
    1. \( y = \left \lvert f\left( x \right) \right \rvert \)
    2. \( y = f\left( \left \lvert x \right \rvert \right) \)